Estimation of Superpopulation Parameters
187
( )
∂
=
−
−
∂
2
2
2
log
log
(
)
f y
y m
s
s
m
(A.30)
Defi ne 0
1
ˆ
log
n
j
i
x n
=
= ∑
m
and ( )
( )
=
=
−
∑
2
2
1
ˆ
( log
)
. Let { }
n
v
j
j
x
n
s m
m
denote the current estimate of . Then the EM iteration
(v)
→
(v11) from
Equation A.28 is given as
( )
( )
( )
+
+
=
+ −
−
1
1 2
0
ˆ
1
E log
,
(
)
v
v
v
n
n
n
A
N
N
x
m
m
m
(A.31)
( )
( )
( )
( )
+
+
+
=
+ −
−
2 1
1
1
2
2
ˆ
1
E log
,
(
)
(
)
v
v
v
v
n
n
n
A
N
N
x
s
s m
m
(A.32)
Given {
(v)
}, we compute μ
(v11) and substitute it into Equation A.32
to get the next estimate of s
2 , and repeat this procedure until either the
log-likelihood function of Equation A.11 stops improving or the absolute difference between {
(v)
} and
(v11) is suffi ciently small. To carry out
this program, we need to calculate the conditional expectations.
Let ' 5 (μ', s'). Then the conditional expectations of log A and (log
A − μ)
2 given the data x n and ' are
∞
=
∫ 0
E log
, '
E log
, '
, '
(
)
n
n
A
A
d
x
x
l
j l
l
(A.33)
2
2
0
E log
, '
E log
, '
, '
(
)
(
)
(
)
n
n
A
A
d
∞
−
=
−
∫
x
x
m
m l
j l
l
(A.34)
where j (•|x n , ' ) is given by Equation A.19 and the conditional expectations inside the integrals are taken with respect to Equation A.22.
Further manipulation, by using Equations A.25, A.29, and A.30,
yields
2 '
E log
, '
'
log
'
'
(
)
f
A
∂
= +
∂
l
m s
r l
m
(A.35)
(
)
2
2
2
E log
, '
'
' 1 2 '
log
'
'
log
'
'
'
(
)
(
)
(
)
(
)
f
f
A
−
=
−
∂
∂
+
+
−
+
∂
∂
m l
m m
s
m m
r l
s
r l
m
s
(A.36)
187
( )
∂
=
−
−
∂
2
2
2
log
log
(
)
f y
y m
s
s
m
(A.30)
Defi ne 0
1
ˆ
log
n
j
i
x n
=
= ∑
m
and ( )
( )
=
=
−
∑
2
2
1
ˆ
( log
)
. Let { }
n
v
j
j
x
n
s m
m
denote the current estimate of . Then the EM iteration
(v)
→
(v11) from
Equation A.28 is given as
( )
( )
( )
+
+
=
+ −
−
1
1 2
0
ˆ
1
E log
,
(
)
v
v
v
n
n
n
A
N
N
x
m
m
m
(A.31)
( )
( )
( )
( )
+
+
+
=
+ −
−
2 1
1
1
2
2
ˆ
1
E log
,
(
)
(
)
v
v
v
v
n
n
n
A
N
N
x
s
s m
m
(A.32)
Given {
(v)
}, we compute μ
(v11) and substitute it into Equation A.32
to get the next estimate of s
2 , and repeat this procedure until either the
log-likelihood function of Equation A.11 stops improving or the absolute difference between {
(v)
} and
(v11) is suffi ciently small. To carry out
this program, we need to calculate the conditional expectations.
Let ' 5 (μ', s'). Then the conditional expectations of log A and (log
A − μ)
2 given the data x n and ' are
∞
=
∫ 0
E log
, '
E log
, '
, '
(
)
n
n
A
A
d
x
x
l
j l
l
(A.33)
2
2
0
E log
, '
E log
, '
, '
(
)
(
)
(
)
n
n
A
A
d
∞
−
=
−
∫
x
x
m
m l
j l
l
(A.34)
where j (•|x n , ' ) is given by Equation A.19 and the conditional expectations inside the integrals are taken with respect to Equation A.22.
Further manipulation, by using Equations A.25, A.29, and A.30,
yields
2 '
E log
, '
'
log
'
'
(
)
f
A
∂
= +
∂
l
m s
r l
m
(A.35)
(
)
2
2
2
E log
, '
'
' 1 2 '
log
'
'
log
'
'
'
(
)
(
)
(
)
(
)
f
f
A
−
=
−
∂
∂
+
+
−
+
∂
∂
m l
m m
s
m m
r l
s
r l
m
s
(A.36)